Nature of mathematics

This work develops and defends a structural view of the nature of mathematics,
which is used to explain a number of striking features of mathematics
that have puzzled philosophers for centuries. It rejects the most widely
held philosophical view of mathematics (Platonism), according to which
mathematics is a science dealing with mathematical objects such as sets and
numbers—objects which are believed not to exist in the physical world.

Annals of Mathematics
This paper is the third in a series where we describe the space of all embedded minimal surfaces of ﬁxed genus in a ﬁxed (but arbitrary) closed 3-manifold. In [CM3]–[CM5] we describe the case where the surfaces are topologically disks on any ﬁxed small scale. Although the focus of this paper, general planar domains, is more in line with [CM6], we will prove a result here (namely, Corollary III.

The ﬁrst is the widely held view that mathematics is, somehow, innate11.
Pre-service teachers will often indicate that they do not see the need to learn
the material being covered because, when the time comes that they actually
need it, they will be able to dredge it up.

Annals of Mathematics
In our ﬁrst article [2] we developed a new view of Gauss composition of binary quadratic forms which led to several new laws of composition on various other spaces of forms. Moreover, we showed that the groups arising from these composition laws were closely related to the class groups of orders in quadratic number ﬁelds, while the spaces underlying those composition laws were closely related to certain exceptional Lie groups.

Annals of Mathematics
By S. Artstein, V. Milman, and S. J. Szarek
For two convex bodies K and T in Rn , the covering number of K by T , denoted N (K, T ), is deﬁned as the minimal number of translates of T needed to cover K. Let us denote by K ◦ the polar body of K and by D the euclidean unit ball in Rn . We prove that the two functions of t, N (K, tD) and N (D, tK ◦ ), are equivalent in the appropriate sense, uniformly over symmetric convex bodies K ⊂...

We study a 1D transport equation with nonlocal velocity and show the formation of singularities in ﬁnite time for a generic family of initial data. By adding a diﬀusion term the ﬁnite time singularity is prevented and the solutions exist globally in time. 1. Introduction In this paper we study the nature of the solutions to the following class of equations (1.1)

We prove that if f (x) = n−1 ak xk is a polynomial with no cyclotomic k=0 factors whose coeﬃcients satisfy ak ≡ 1 mod 2 for 0 ≤ k 1 + log 3 , 2n
resolving a conjecture of Schinzel and Zassenhaus [21] for this class of polynomials. More generally, we solve the problems of Lehmer and Schinzel and Zassenhaus for the class of polynomials

Annals of Mathematics
We study the motion of an incompressible perfect liquid body in vacuum. This can be thought of as a model for the motion of the ocean or a star. The free surface moves with the velocity of the liquid and the pressure vanishes on the free surface. This leads to a free boundary problem for Euler’s equations, where the regularity of the boundary enters to highest order. We prove local existence in Sobolev spaces assuming a “physical condition”, related to the fact that the pressure of a ﬂuid has to be positive. ...

Annals of Mathematics
In this paper we will solve one of the central problems in dynamical systems: Theorem 1 (Density of hyperbolicity for real polynomials). Any real polynomial can be approximated by hyperbolic real polynomials of the same degree. Here we say that a real polynomial is hyperbolic or Axiom A, if the real line is the union of a repelling hyperbolic set, the basin of hyperbolic attracting periodic points and the basin of inﬁnity.

That is all: just a computer procedure to approximate a real root. From
the narrow perspective of treating mathematics as a tool to solve real life
problems, this is of course suﬃcient. However, from the point of view of
mathematics, shouldn’t a student be interested in roots of polynomials in
general? Fourth degree? Odd degree? Other roots, once one is found?
Rational roots? Total number of roots?
Not every detail need be explained, but even the average student will
have his life improved by the mere knowledge that there are such questions,
often with answers, e.g.

Like all other sciences, physics is based on experimental observations and quantitative measurements. The main objective of physics is to ﬁnd the limited number of fundamental laws that govern natural phenomena and to use them to develop theories that can predict the results of future experiments. The fundamental laws used in developing theories are expressed in the language of mathematics, the tool that provides a bridge between theory and experiment

Stochastic Calculus of Variations (or Malliavin Calculus) consists, in brief,
in constructing and exploiting natural differentiable structures on abstract
probability spaces; in other words, Stochastic Calculus of Variations proceeds
from a merging of differential calculus and probability theory.
As optimization under a random environment is at the heart of mathematical
finance, and as differential calculus is of paramount importance for the
search of extrema, it is not surprising that Stochastic Calculus of Variations
appears in mathematical finance.

We verify an old conjecture of G. P´lya and G. Szeg˝ saying that the o o regular n-gon minimizes the logarithmic capacity among all n-gons with a ﬁxed area. 1. Introduction The logarithmic capacity cap E of a compact set E in R2 , which we identify with the complex plane C, is deﬁned by (1.1) − log cap E = lim (g(z, ∞) − log |z|),
z→∞
where g(z, ∞) denotes the Green function of a connected component Ω(E) ∞ of C \ E having singularity at z = ∞; see [4, Ch. 7], [7, §11.1]. By an n-gon with...

We completely classify diffeomorphism covariant local nets of von Neumann
algebras on the circle with central charge c less than 1. The irreducible
ones are in bijective correspondence with the pairs of A-D2n-E6,8 Dynkin diagrams
such that the difference of their Coxeter numbers is equal to 1.
We first identify the nets generated by irreducible representations of the
Virasoro algebra for c

We study “ﬂat knot types” of geodesics on compact surfaces M 2 . For every ﬂat knot type and any Riemannian metric g we introduce a Conley index associated with the curve shortening ﬂow on the space of immersed curves on M 2 . We conclude existence of closed geodesics with prescribed ﬂat knot types, provided the associated Conley index is nontrivial. 1. Introduction If M is a surface with a Riemannian metric g then closed geodesics on (M, g) are critical points of the length functional L(γ) = |γ (x)|dx deﬁned on the space of unparametrized C...

Suppose that G is a locally compact abelian group, and write M(G) for the algebra of bounded, regular, complex-valued measures under convolution. A measure µ ∈ M(G) is said to be idempotent if µ ∗ µ = µ, or alternatively if µ takes only the values 0 and 1. The Cohen-Helson-Rudin idempotent theorem states that a measure µ is idempotent if and only if the set {γ ∈ G : µ(γ) = 1} belongs to the coset ring of G, 1. Introduction Let

We show that unital simple C ∗ -algebras with tracial topological rank zero which are locally approximated by subhomogeneous C ∗ -algebras can be classiﬁed by their ordered K-theory. We apply this classiﬁcation result to show that certain simple crossed products are isomorphic if they have the same ordered K-theory. In particular, irrational higher dimensional noncommutative tori of the form C(Tk ) ×θ Z are in fact inductive limits of circle algebras. Introduction In recent years there has been rapid progress in classiﬁcation of nuclear simple C ...